Lesson 1: Classifying Triangles by Sides and Angles

Property

To determine if a triangle is possible at all, check the Side Length Condition (Triangle Inequality): The sum of the lengths of any two sides must be strictly greater than the length of the third side. If the sum of the two shorter sides is less than or equal to the longest side, no triangle is formed.

Examples

• Side lengths of 2 cm, 3 cm, and 7 cm cannot form a triangle because 2 + 3 < 7. The two shorter sides are not long enough to meet.
• Given side lengths of 3, 4, and 8, since 3 + 4 is less than or equal to 8, the two shorter sides are not long enough to connect, so no triangle is formed.

Explanation

When constructing a triangle, the given measurements act as a set of instructions. If the side lengths are too short to connect, no triangle can be formed. The two smaller sides combined must always be longer than the biggest side, otherwise, they will just collapse flat!

Property

Triangles can be classified by the size of their angles:

• An acute triangle has all three angles measuring less than $90^\circ$.
• A right triangle has one angle measuring exactly $90^\circ$.
• An obtuse triangle has one angle measuring greater than $90^\circ$.
• An equiangular triangle has all three angles measuring exactly $60^\circ$.

Examples

• A triangle with angles measuring $50^\circ, 60^\circ, 70^\circ$ is an acute triangle.
• A triangle with angles measuring $30^\circ, 60^\circ, 90^\circ$ is a right triangle.
• A triangle with angles measuring $40^\circ, 30^\circ, 110^\circ$ is an obtuse triangle.
• A triangle with angles measuring $60^\circ, 60^\circ, 60^\circ$ is an equiangular triangle.

Explanation

Triangles can be classified based on the measures of their interior angles. If all three angles are acute (less than $90^\circ$), the triangle is an acute triangle. If one angle is a right angle (exactly $90^\circ$), it is a right triangle. If one angle is obtuse (greater than $90^\circ$), it is an obtuse triangle.

Property

A triangle can be classified by both its side lengths and its angle measures. This gives a more precise description of the triangle, using one term for its sides and one for its angles.

Examples

• A triangle with side lengths $3, 4, 5$ and angles $90^\circ, 53^\circ, 37^\circ$ is a right scalene triangle.
• A triangle with side lengths $7, 7, 10$ and angles $45^\circ, 45^\circ, 90^\circ$ is an right isosceles triangle.
• A triangle with side lengths $8, 8, 8$ and angles $60^\circ, 60^\circ, 60^\circ$ is an equiangular equilateral triangle (or acute equilateral triangle).

Explanation

Every triangle has two names that describe its properties. One name is based on the lengths of its sides (equilateral, isosceles, or scalene). The other name is based on the measures of its angles (acute, right, obtuse, or equiangular). By combining these two classifications, we can provide a more complete and specific description for any triangle.